In a bag, the ratio of the number of blue balls to the number of red balls is constant. When there were 44 red balls in the bag, the number of blue balls was 36. If later the number of blue balls becomes 54 while keeping the same ratio, how many red balls will be in the bag?

Introduction / Context:
This question checks understanding of ratios and proportional reasoning. The ratio between two types of objects is kept constant, and we use the initial information to find the new quantity of one type when the quantity of the other type changes. This is a common type of question in aptitude tests and forms a basic application of ratios and proportions.

Given Data / Assumptions:

• The ratio of blue balls to red balls remains constant at all times.
• When there are 44 red balls, there are 36 blue balls.
• Later, the number of blue balls becomes 54.
• We need to find the corresponding number of red balls when the ratio is unchanged.

Concept / Approach:
When the ratio between two quantities is constant, they are directly proportional. This means blue / red = constant. We can first compute the ratio using the initial data, then set up a proportion involving the new number of blue balls and the unknown number of red balls. Solving the resulting proportion yields the required value.

Step-by-Step Solution:
Step 1: Compute the initial ratio of blue to red balls using 36 blue and 44 red.Step 2: Simplify the ratio 36 : 44 by dividing both numbers by their greatest common divisor, which is 4, giving 9 : 11.Step 3: Let the new number of red balls be R when the number of blue balls is 54. Since the ratio stays 9 : 11, we have 54 : R = 9 : 11.Step 4: Use the property of proportions: 54 / R = 9 / 11, so R = 54 × 11 / 9.Step 5: Calculate R: first 54 / 9 = 6, then 6 × 11 = 66.Step 6: Therefore, the number of red balls needed to maintain the same ratio is 66.
Verification / Alternative check:
We can verify by checking the ratio again with the new numbers: 54 blue and 66 red. The simplified ratio is 54 : 66. Divide both by 6 to get 9 : 11, which is exactly the original ratio. Hence the solution is consistent with the requirement that the ratio remains constant.

Why Other Options Are Wrong:
Values such as 64, 68, 32 or 72 do not give the ratio 9 : 11 when paired with 54. For example, 54 : 64 simplifies approximately to 27 : 32, which is not the same as 9 : 11. Therefore these options violate the condition of a constant ratio and must be rejected.

Common Pitfalls:
Many students mistakenly set up the ratio in the reverse order or misapply cross multiplication. Some also forget to simplify the ratio first and try to guess the answer. Systematically writing the proportion and solving it avoids such errors and ensures accuracy in similar ratio based questions.

Final Answer:
The number of red balls in the bag when there are 54 blue balls is 66.